The Atmospheric Kinetic Energy Spectrum and Nonlinear ...aerler/files/...Total Kinetic Energy spectra; -3.1 and-2.5 slopes have been added. Andre R. Erler, A. B. Helen Burgess, Theodore

Large-scale KE SpectrumMeso-scale Shallowing of KE Spectrum

Nonlinear Spectral Fluxes

The Atmospheric Kinetic Energy Spectrum andNonlinear Spectral Fluxes as seen in ECMWF

Operational Analyses

Andre R. Erler A. B. Helen Burgess Theodore G. Shepherd

University of Toronto

July 10th, 2013

Andre R. Erler, A. B. Helen Burgess, Theodore G. Shepherd The Kinetic Energy Spectrum in ECMWF Operational Analysis



Outline

The Large-scale Kinetic Energy SpectrumIntroduction: Synoptic to Planetary ScalesSynoptic to Planetary Scales in ECMWF Analysis

Meso-scale Shallowing of the Kinetic Energy SpectrumIntroduction: The Mesoscale KE SpectrumMeso-scale Shallowing and Change-point AnalysisThe Divergent Component and Gravity Waves

Nonlinear Spectral FluxesTurbulence at the TropopauseEddy / Mean–flow Interaction in the Stratosphere



Nonlinear Spectral FluxesIntroduction: Synoptic to Planetary ScalesSynoptic to Planetary Scales in ECMWF Analysis

Outline







Atmospheric Turbulence at Large Scales[SW06]

Eddy Kinetic Energy spectrum fromERA-40 reanalysis. Large scales areforced at the scale of the Rossby radius;and any inverse cascade would be haltedat the Rhines scale.

In the atmosphere macro-turbulence is forced bysynoptic-scale baroclinicinstability. A direct enstrophycascade exists, but no inverseenergy cascade.

No Inverse CascadeBaroclinic eddies always ad-just the stratification suchthat the Rhines scale coin-cides with the scale of ba-roclinic instability.




The Kinetic Energy Spectrum at Large Scales

Boer & Shepherd (1983)first computed KE spectrafor meteorological analysisdata (T31), showingresemblance to 2Dgeostrophic turbulence.

However, a strongstationary zonal flow alsoexists. 2D turbulence–likeflow is limited to transientcomponent.

[BS83]

Kinetic energy components for:a) stationary/transient,b) zonal/meridional, andc) zonal/meridional transient.




ECMWF Analysis DataModel

I IFS (Integrated ForecastSystem) with dataassimilation (4DVar)

I global spectral model,run at T799 resolution

I hydrostatic with hybridvertical coordinate

Data

I Gridded data:0.25◦ × 0.25◦ (∼ 25 km),91 vertical levels

I provided in gridded form byECMWF for IPY (2008/09)

I we use one month of data(January 2008)

Note: All spectra are in total spherical harmonic wavenumber




ECMWF Analysis DataModel

I IFS (Integrated ForecastSystem) with dataassimilation (4DVar)

I global spectral model,run at T799 resolution

I hydrostatic with hybridvertical coordinate

Data

I Gridded data:0.25◦ × 0.25◦ (∼ 25 km),91 vertical levels

I provided in gridded form byECMWF for IPY (2008/09)

I we use one month of data(January 2008)

Note: All spectra are in total spherical harmonic wavenumber




The Kinetic Energy Spectrum in ECMWF Analysis

Total Kinetic Energy

I Steep power law insynoptic scales

I Meso-scale shallowingin stratosphere

I Planetary scales quite“noisy”

I Strong dissipationbeyond n ≈ 200

Total Kinetic Energy spectra; -3.1 and-2.5 slopes have been added.




The Stationary and Transient Components

1 10 100n

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Kin

eti

c Energ

y [

m2

s-2]

(a)

StationaryTransientTotal, 250 hPa

1 10 100n

(b)

StationaryTransientTotal, 10 hPa

1 10 100n

(c)

500 hPa250 hPa100 hPa 50 hPa 25 hPa 10 hPa

Stationary and Transient KE spectra at 250 hPa (a) and 10 hPa(b), and transient KE at several levels (c).




Zonal-mean Spectrum and Anisotropy

1 10 100n

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Kin

eti

c Energ

y [

m2

s-2]

(a)

Zonal, 250 hPaMeridional

1 10 100n

(b)

Zonal, 10 hPaMeridional

0.5 1.0 1.5 2.0 2.5Anisotropy

10

100

20

40

200

400

p

[hPa]

(c)

CP 1CP 2

Zonal and meridional components of the transient flow at 250 hPa(a) and 10 hPa (b), and anisotropy at the synoptic and meso-scale.




Summary: Large-scale Spectra

I Synoptic scale slope ≈ −3, steepening with altitudeI Meso-scale shallowing emerging in the tropopause regionI Strong dissipation beyond n ≈ 200

I Planetary scales dominated by stationary flowI Transients dominate synoptic and smaller scales

I Anisotropy of is relatively large in synoptic scale (jets)I Much stronger anisotropy in the stratosphere




Introduction: The Mesoscale KE SpectrumMeso-scale ShallowingThe Divergent Component

Outline








Observations: the Nastrom & Gage–Spectrum

Wind and Temperaturemeasurements from commercialaircraft.Flight levels in tropopause region.

Horizontal resolution: 2.5 km

I k−5/3-spectrum in mesoscale upto ∼400 km

I k−3-spectrum at large scalesI relatively isotropic

N.B.: these data are much higherresolution than any GCM, even now!

[NG85]

Zonal and meridional wind andtemperature (offset by onedecade each).





Observations: the Nastrom & Gage–Spectrum

Wind and Temperaturemeasurements from commercialaircraft.Flight levels in tropopause region.

Horizontal resolution: 2.5 km

I k−5/3-spectrum in mesoscale upto ∼400 km

I k−3-spectrum at large scalesI relatively isotropic

N.B.: these data are much higherresolution than any GCM, even now!

[NG85]

Zonal and meridional wind andtemperature (offset by onedecade each).





High Resolution GCM Simulations

From Takahashi et Al.2006, using the EarthSimulator at T639.

I The first mesoscaledecade is wellresolved in the model.

I The mesoscalespectral shallowing ismore pronouncedthan in observations.

Wind spectra from the model andaircraft measurements [THO06]





Controversy over the Meso–scale KE Spectrum I

The nature and forcing of the mesoscale cascade are stillcontroversial, but most recent evidence points towards a directcascade.

Direct CascadeI VanZandt 1982I Lindborg 1999I Koshyk & Hamilton 2001

Unbalanced motion (e.g. gravitywaves) excited by, e.g., baroclinicwaves. Cascading down fromlarger to smaller scales.

Inverse CascadeI Lilly 1983I Vallis, Shutts, & Gray 1997

Balanced motion forced fromsmall scales, e.g. by convection.Energy cascades from small tolarge scales.





Meso–scale Shallowing of the Kinetic Energy Spectrum

In the tropopause regionthe spectrum develops twodistinct wavenumberregimes, which remain inthe stratosphere.

I Spectral shallowingbetween 250 hPa and100 hPa

I Shallowing starts atlarger scales at higheraltitudes Total Kinetic Energy spectra; -3.1 and

-2.5 slopes have been added.





Meso–scale Shallowing of the Kinetic Energy Spectrum

In the tropopause regionthe spectrum develops twodistinct wavenumberregimes, which remain inthe stratosphere.

I Spectral shallowingbetween 250 hPa and100 hPa

I Shallowing starts atlarger scales at higheraltitudes Total KE spectra between 250 hPa and

100 hPa; -3.1 and -2.5 slopes added.





Change–point Analysis

Piece–wise linear fit with freeparameter H:

I y(x) = y0 + γ1(x − x0)for x < H

I y(x) = yH + γ2(x − H)for x > H

with yH = y0 + γ1(H − x0)

H and γ1,2 are adjusted to min-imize Root–Mean–Square Er-ror of the fit against the profile

sample function (blue)





Change–point Analysis

Piece–wise linear fit with freeparameter H:

I y(x) = y0 + γ1(x − x0)for x < H

I y(x) = yH + γ2(x − H)for x > H

with yH = y0 + γ1(H − x0)

H and γ1,2 are adjusted to min-imize Root–Mean–Square Er-ror of the fit against the profile

sample function (blue)fitted function (green)





Meso-scale ShallowingUse change–point analysisto fit a two–segment linearfunction to the KEspectrum.

I Estimate transitionwavenumber ncp,

I and synoptic andmeso–scale slopes

The fit is robust; but ifthere is no shallowing, ncpis not meaningful.

Piece–wise linear fits to KE spectrum.





Meso-scale ShallowingUse change–point analysisto fit a two–segment linearfunction to the KEspectrum.

I Estimate transitionwavenumber ncp,

I and synoptic andmeso–scale slopes

The fit is robust; but ifthere is no shallowing, ncpis not meaningful.

Profile of change-point wavenumberand slopes.





Controversy over the Meso–scale KE Spectrum II

Currently the inverse cascade hypothesis is out of favor, but forcingand dynamics are still unclear.

Gravity Waves

among others:I VanZandt 1982I Koshyk & Hamilton 2001

Unbalanced motion excited by,e.g., baroclinic waves or con-vection. Associated with di-vergent component.

Stratified Turbulence

among others:I Lindborg 2007I Hakim et Al. 2002

Balanced motion forced, e.g.by synoptic scales or convec-tion. Associated with rota-tional component.





The Rotational and Divergent Components

Divergent and Rotationalcomponents at 250 hPa.

I Rotationalcomponent representsbalanced flow

I Divergent componentrepresents unbalancedflow

Divergent KE

Meso–scale shallowing isdriven by divergent KE.





The Rotational and Divergent Components

Divergent and Rotationalcomponents at 10 hPa.

I Rotationalcomponent representsbalanced flow

I Divergent componentrepresents unbalancedflow

Divergent KE

Meso–scale shallowing isdriven by divergent KE.





The Divergent Component and Meso-scale Shallowing

Blow-up of divergent androtational components atdifferent levels, and theirintersection points (opencircles).

Intersection PointA clear progression ofthe intersection point tolarger scales with alti-tude is evident.





The Divergent Component and Meso-scale Shallowing

Wavenumber at whichrotational and divergentcomponents intersect as afunction of height.Change-point wavenumberis also plotted.

I Intersection andchange-point followthe same pattern

I Note: divergent KEbecomes significantbefore intersection





Rotational andDivergent KE

I (a, b)Rotational andDivergentcomponents

I (c) Intersectionof components

I (d)Wavenumberwhere div = roand ncp

10-5

10-4

10-3

10-2

10-1

100

101

102

Kin

eti

c Energ

y [

m^

2 s

^-2

]

(a)

Div., 250 hPaRot., 250 hPaTot., 250 hPa

10-3

10-2

10-1

100

Kin

eti

c Energ

y [

m^

2 s

^-2

]

(c)

Div., 100 hPaRot., 100 hPaDiv., 150 hPaRot., 150 hPaDiv., 200 hPaRot., 200 hPaDiv., 250 hPaRot., 250 hPa

1 10 100n

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Kin

eti

c Energ

y [

m^

2 s

^-2

]

(b)


10 1004 6 20 40 60 200 400n

10

100

20

40

60

200

400

600

900

p [

hPa]

(d)

div = rotflux = 0n_cp





Meso-scale Shallowing and Divergent KE

Slope of the divergent KEspectrum (purple).

Hypothesis

In the absence of dis-sipation the meso-scaleslope may converge to-wards the observed slopeof divergent KE.

R: Profiles of change- pointwavenumber and slopes.





Density-weighted Spectra

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Div

erg

ent

KE

[m^

2 s

^-2

]

400 hPa

200 hPa

100 hPa

50 hPa

25 hPa

1 10 100n

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Rota

tional K

E

[m^

2 s

^-2

]

400 hPa

200 hPa

100 hPa

50 hPa

25 hPa

1 10 100n

Divergent (top) and rotational (bottom)KE components.Left: Normal KE spectrumRight: Density–scaled spectrum

Divergent motion isassociated withunbalanced flow; therotational component withbalanced motion.

I “Normal” KE spectraare actually velocityvariance spectra

I Scaling by densityyields actual kineticenergy





Gravity Wave Propagation in the Stratosphere

I Divergent KE isalmost constant inthe troposphere

I Drops most between100 hPa and 50 hPa

I Rotational KE dropsoff rapidly

Gravity Waves

Consistent with up-wards propagating (in-ertia) gravity waves.

100n

10-6

10-5

10-4

10-3

10-2

10-1

100

101

densi

ty-s

cale

d K

ineti

c Energ

y p

er

Volu

me [

J /

m^

3]

Divergent KE

400 hPa

200 hPa

100 hPa

50 hPa

25 hPa

100n

Rotational KE

Closeup of density–scaled divergent(left) and rotational (right) components





Gravity Wave Propagation in the Stratosphere

I Divergent KE isalmost constant inthe troposphere

I Drops most between100 hPa and 50 hPa

I Rotational KE dropsoff rapidly

Gravity Waves

Consistent with up-wards propagating (in-ertia) gravity waves.

100n

10-6

10-5

10-4

10-3

10-2

10-1

100

101

densi

ty-s

cale

d K

ineti

c Energ

y p

er

Volu

me [

J /

m^

3]

Divergent KE

400 hPa

200 hPa

100 hPa

50 hPa

25 hPa

100n

Rotational KE

Closeup of density–scaled divergent(left) and rotational (right) components





Gravity Wave Breaking in the Stratosphere

Relative gradient of (densityweighted) kinetic energycomponents:

I Rotational (top):decrease in uppertroposphere(Charney-DrazinFiltering)

I Divergent (bottom):decrease in stratosphere(gravity wave breaking)

10

100

20

50

200

500

Pre

ssure

[hPa]

10 10020 50 200Total wavenumber n

10

100

20

50

200

500

Pre

ssure

[hPa]

2.5

1.0

0.5

2.0

3.5

5.0

2.5

1.0

0.5

2.0

3.5

5.0





Summary: Meso-scale Shallowing

I A shallow meso-scale range emerges at tropopause andpersists into the stratosphere

I With increasing altitude the change-point wavenumber movesto larger scales

I Shallowing is mainly associated with the divergent componentI The slope of the divergent component is close to −5

3

I The rotational component is preferentially damped withaltitude, allowing the divergent component to emerge.

I Hypothesis: the shallow meso–scale spectrum may be causedby gravity waves propagating into the stratosphere




Turbulence at the TropopauseEddy / Mean–flow Interaction in the StratosphereSummary & Conclusion

Outline








Nonlinear Triad Interactions

I Consider tendencies due to advection term in wavenumberspace

∂u∂t = u · ∇u , u =

∑k

uke2πi k·x

∑k

e2πi k·x ∂uk∂t =

∑m,l

umul e2πi (m+l)·x

I By orthogonality of the basis functions, the wavenumbershave to add up:

k = m + l





Rotational KineticEnergy Fluxes

KE fluxes are theintegral of thenonlinear interactionterms (rotationalcomponent only).

I Nonlinear fluxesare much strongerin the troposphere

I Synoptic sourceregion

Nonlinear interaction terms (top) and fluxes (bottom)in the troposphere (left) and stratosphere (right).





Rotational KineticEnergy Fluxes

Troposphere (left):I Upscale KE flux in

synoptic scalesI Downscale KE flux

in meso-scaleStratosphere (right):

I Upscale KE flux toplanetary scale

I Meso-scale KEflux is very small

Large-scale (top) and meso-scale (bottom) nonlinearfluxes in the troposphere (left) and stratosphere(right).





Rotational Kinetic Energy Fluxes

Two centers of inverseKE flux emerge in thetropopause region:

I at large synopticscales (n = 10)

I and small planetaryscales (n = 5)

2D TurbulenceThe upscale KE flux islimited to large scales atthe tropopause

1 10 1002 5 20 50Total wavenumber n

100

1000

50

200

500

Pre

ssure

[hPa]

10

8

6

4

2

0

2

Kinetic energy fluxes resulting fromnonlinear interactions of the rotationalcomponent only (quasi–2D)





Rotational Kinetic Energy Fluxes

Two centers of inverseKE flux emerge in thetropopause region:

I at large synopticscales (n = 10)

I and small planetaryscales (n = 5)

2D TurbulenceThe upscale KE flux islimited to large scales atthe tropopause

1 10 1002 5 20 50Total wavenumber n

100

1000

50

200

500

Pre

ssure

[hPa]

10

8

6

4

2

0

2

Kinetic energy fluxes resulting fromnonlinear interactions of the rotationalcomponent only (quasi–2D)





Eddy / Mean–flow Interaction

Total, Eddy / Eddy, and Eddy /Mean–flow interactions (of therotational component) at 250 hPa

Troposphere (250 hPa)

I Eddy / Eddy interactionsdominate in synoptic scales

I Eddy / Mean–flowinteractions dominateplanetary scale

Upscale KE Transfer

The inverse KE cascade is ahand–off from Eddy / Eddy toEddy / Mean–flow interaction.







Troposphere (250 hPa)

I Eddy / Eddy interactionsdominate in synoptic scales

I Eddy / Mean–flowinteractions dominateplanetary scale

Upscale KE Transfer

The inverse KE cascade is ahand–off from Eddy / Eddy toEddy / Mean–flow interaction.







Stratosphere (10 hPa)

I Nonlinear interactions areweaker (∼ 1

3 )I Eddy / Mean–flow

Interactions dominate evenmore

Wave / Mean–flow

Eddy / Mean–flow interactionprobably mostly due to waves,consistent with E-P theory.







Stratosphere (10 hPa)

I Nonlinear interactions areweaker (∼ 1

3 )I Eddy / Mean–flow

Interactions dominate evenmore

Wave / Mean–flow

Eddy / Mean–flow interactionprobably mostly due to waves,consistent with E-P theory.





Eddy / Mean–flowInteractionand 2D Turbulence

The tropospherehas a turbulentenstrophy cascade

In the stratospherethe enstrophycascade is weak

Right: KE (top) andenstrophy (bottom)flux at 250 hPa (left)and 10 hPa (right)

8

6

4

2

0

2

KE F

lux [

10

-4m

3s-3

]

(a)

Total, 250 hPaZonal-meanEddy-eddy

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0.5

(c)


1 10 1002 5 20 50 200n

0.5

0.0

0.5

1.0

1.5

2.0 E

nst

rophy F

lux [

10

-15

s-3]

(b)


1 10 1002 5 20 50 200n

0.02

0.00

0.02

0.04

0.06

0.08

0.10

(d)






Summary: Nonlinear Fluxes

I Strong 2D-turbulence–like fluxes in the tropopause region:upscale KE flux and downscale enstrophy flux from synopticsource

I Meso-scale KE flux is downscale, but relatively weak

I Nonlinear Eddy / Eddy interactions in the stratosphere areweak

I Energy transfer to large planetary scales is achieved byEddy / Mean-flow interactions, in both stratosphere andtroposphere





Comparison with a Higher Resolution Forecast Model

1 10 100 1000n

10-5

10-4

10-3

10-2

10-1

100

101

102

Kin

eti

c Energ

y [

m2

s-2]

(a)

T721 IPY Data, 250 hPaT799 Analysis, 250 hPaT1279 Forecast, 250 hPa

10 100 1000n

(b)

T721 IPY Data, 100 hPaT799 Analysis, 100 hPaT1279 Forecast, 100 hPa

Total Kinetic Energy spectra for the N721 interpolated (IPY)analysis, the T799 analysis, and a T1279 forecast at 250 hPa (a)and 100 hPa (b); slopes of -3 and −5

3 have been added.





Summary

I The large–scale KE spectrum is reproduced in the ECMWFanalysis data, but the meso–scale spectrum is too steep

I Meso–scale Shallowing occurs very suddenly in the tropopauseregion and persists throughout the stratosphere

I The shallowing is driven by divergent KE; there is someevidence for upward propagating (inertia) gravity waves

I Strong turbulence occurs predominantly in the tropopauseregion (direct and indirect cascade)

I Upscale KE flux is taken over by Eddy / Mean–flow Interactionat the planetary scale (arrest of the inverse cascade)





Summary










Summary










Caveats

I All spectra and fluxes are global, hence averaged over tropicaltroposphere and extra-tropical stratosphere

I Comparison with aircraft measurements is difficult becausethese are hemispherically biased and averaged over thetroposphere

I Strong dissipation beyond n ≈ 300, downscale fluxes not fullyconverged





References I[BS83] G. Boer and T. Shepherd. Large scale 2-d turbulence in the atmosphere. J.

Atmos. Sci, 40:164–184, 1983.[Cha71] J G Charney. Geostrophic turbulence. J. Atmos. Sci., 28(6):1087, 1971.

[HSM02] G J Hakim, C Snyder, and D J Muraki. A new surface model forcyclone–anticyclone asymmetry. J. Atmos. Sci., 59(16):2405–2420, AUG2002.

[HTO08] Kevin Hamilton, Yoshiyuki O. Takahashi, and Wataru Ohfuchi. Mesoscalespectrum of atmospheric motions investigated in a very fine resolutionglobal general circulation model. J. Geophys. Res, 113(D18):1–19,September 2008.

[KH01] J.N. Koshyk and K. Hamilton. The horizontal kinetic energy spectrum andspectral budget simulated by a high-resolutiontroposphere-stratosphere-mesosphere gcm. Journal of the atmosphericsciences, 58(4):329–348, 2001.

[LB07] E. Lindborg and G. Brethouwer. Stratified turbulence forced in rotationaland divergent modes. Journal of Fluid Mechanics, 586:83, 2007.

[Lin99] E. Lindborg. Can the atmospheric kinetic energy spectrum be explained bytwo-dimensional turbulence? Journal of Fluid Mechanics, 388:259–288,1999.





References II

[NG85] G D Nastrom and K S Gage. A climatology of atmospheric wavenumberspectra of wind and temperature observed by commercial aircraft. J.Atmos. Sci., 42(9):950–960, 1985.

[SD99] D.M. Straus and P. Ditlevsen. Two-dimensional turbulence properties of theecmwf reanalyses. Tellus A, 51(5):749–772, 1999.

[She87] T G Shepherd. A spectral view of nonlinear fluxes and stationary transientinteraction in the atmosphere. J. Atmos. Sci., 44(8):1166–1178, APR 1987.

[SW06] T Schneider and C C Walker. Self-organization of atmosphericmacroturbulence into critical states of weak nonlinear eddy-eddyinteractions. J. Atmos. Sci., 63(6):1569–1586, JUN 2006.

[THO06] Y.O. Takahashi, K. Hamilton, and W. Ohfuchi. Explicit global simulation ofthe mesoscale spectrum of atmospheric motions. Geophysical researchletters, 33(12):L12812, 2006.

[Van82] TE VanZandt. A universal spectrum of buoyancy waves in the atmosphere.Geophysical Research Letters, 9:575–578, 1982.





Nonlinear Triad Interactions

I Consider tendencies due to advection term in wavenumberspace

∂u∂t = u · ∇u , u =

∑k

uke2πi k·x

∑k

e2πi k·x ∂uk∂t =

∑m,l

umul e2πi (m+l)·x

I By orthogonality of the basis functions, the wavenumbershave to add up:

k = m + l





Rotational andDivergent KE

I (a, b)Rotational andDivergentcomponents

I (c) Intersectionof components

I (d)Wavenumberwhere div = roand ncp

10-5

10-4

10-3

10-2

10-1

100

101

102

Kin

eti

c Energ

y [

m^

2 s

^-2

]

(a)


10-3

10-2

10-1

100

Kin

eti

c Energ

y [

m^

2 s

^-2

]

(c)

Div., 100 hPaRot., 100 hPaDiv., 150 hPaRot., 150 hPaDiv., 200 hPaRot., 200 hPaDiv., 250 hPaRot., 250 hPa

1 10 100n

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Kin

eti

c Energ

y [

m^

2 s

^-2

]

(b)


10 1004 6 20 40 60 200 400n

10

100

20

40

60

200

400

600

900

p [

hPa]

(d)

div = rotflux = 0n_cp


Documents

The Atmospheric Kinetic Energy Spectrum and Nonlinear ...aerler/files/...Total Kinetic Energy spectra; -3.1 and-2.5 slopes have been added. Andre R. Erler, A. B. Helen Burgess, Theodore